Talk:Abstract polytope/Archive 1

Planned content
More precisely, an abstract polytope is a set of objects, supposed to represent the vertices, edges and so on&mdash;the faces&mdash;of the polytope. An "order" is imposed on the set, representing which vert
 * Easy(haha)-to-understand Definition
 * Origin of the study
 * similar concepts
 * geometry, building


 * important research activities / directions
 * amalgamation problem
 * realisations
 * locally toroidal, locally projective

--mike40033 03:09, 22 Dec 2004 (UTC)

I moved the comment about optimisation to the "Polytope" page from here, since it was obviously misplaced. - Cale Gibbard

regular?
Is there any reason this article isn't named abstract regular polytope? It is redirected now. Are there studies of nonregular ones? Tom Ruen 07:27, 12 March 2007 (UTC)


 * Yes, there are. But more attention is paid to the regular abstract polytopes than the non-regular. mike40033 (talk) 04:42, 4 February 2008 (UTC)


 * I notice that we have not yet defined a regular abstract polytope. ISTR that regularity tends to be defined in terms of symmetries within the poset, but I have no handy reference. It seems common these days (and sensible) to talk of "transitivity on the flags" but I do not recall seeing this definition in print. -- Steelpillow (talk) 17:37, 11 April 2008 (UTC)


 * fixed! mike40033 (talk) 08:17, 31 October 2008 (UTC)

How this article could become worthy of the word "article"
The introduction says:

"More precisely, an abstract polytope is an incidence geometry defined on different types of objects, satisfying certain axioms, supposed to represent the vertices, edges and so on — the faces — of the polytope. A linear "order" is imposed on the set of types".

". . . satisying certain axioms . . .", huh??? You may as well just type in the text of Jabberwocky, for all the information that this phase conveys. It is a noble pursuit to initiate a Wikipedia math article, but if your planning hasn't gotten to the point of providing an accurate definition, I would strongly suggest that you hold off on even starting the article.

Surely it wouldn't take that much work to provide the following: 1) one fully accurate definition of "abstract polytope",

2) a clear explanation of how ordinary polytopes are special cases of this, and

3) one example of how this definition applies to an example that's not an ordinary polytope (e.g., the 11-cell).

Then this article would be more than a pile of rubble. Finally it would be terrific to also define -- at least in a linked article -- what an abstract *regular* polytope is.Daqu 20:43, 28 April 2007 (UTC)


 * fixed! mike40033 (talk) 08:18, 31 October 2008 (UTC)


 * Sure it sucks. I just added some pictures, not prepared to edit content myself. Tom Ruen 21:18, 28 April 2007 (UTC)


 * One idea, there's a section on regular forms at least at Regular_polytope. I don't even know if nonregular abstract polytopes have any use. Tom Ruen 22:16, 28 April 2007 (UTC)

call me thick

 * 4. All sections of rank one have a diamond shape

What (if anything) does this mean? &mdash;Tamfang (talk) 22:15, 1 February 2008 (UTC)

A section of Rank 1 is an edge. The (sub)poset has elements 0,a,b,ab and its (Hasse) diagram looks like a square or diamond. Hope that helps. —Preceding unsigned comment added by 81.208.83.208 (talk) 21:12, 12 March 2008 (UTC)


 * fixed now! mike40033 (talk) 08:23, 7 February 2008 (UTC)

Towards a Clear and Concise Axiomatic Definition
(1) It is not necessary to specify that the Poset has a Rank Function. The Rank or Dimensionality of any member of the poset is implied by its "vertical" position in the poset, with the minimal element having Rank = -1. The poset is best illustrated by means of a Hasse diagram, the inversion of which is the dual.


 * To clarify, the key is that the rank function is onto when restricted to a maximal totally ordered subset. mike40033 (talk) 11:59, 11 April 2008 (UTC)

(2) CONNECTEDNESS needs to be defined in this context, which I now do.

Let (P, <) be the poset relation. Then x is an immediate superior of y if y < x and no z satisfies y < z < x. Immediate inferior is defined dually.

Then, x and x' (of equal rank) are ADJACENT (x:x') if they share BOTH an immediate inferior and an immediate inferior. Clearly adjacency is symmetric.

Next, x is CONNECTED to y if there is a finite series a,b,...k such that x:a:b ... k:y. P is k-connected if every pair of k-members is connected.

Finally, the whole structure is connected if it is connected for each rank. In fact it is trivially true that P is both -1 and n-connected, given that there is a minimal and maximal element with rank -1 and n respectively.


 * You are right, there should be a definition of connectedness. The simplest is that any flag can be moved to any other by a sequence of "exchange maps", that is, maps that move a flag to another flag that differs from another by only one element. mike40033 (talk) 11:59, 11 April 2008 (UTC)

(3) I personally feel that this definition is TOO general for many purposes. So I would like to define a NORMAL abstract polytope as one which requires that two distinct k-cells MUST have distinct sets of 0-inferiors and (n-1)-superiors.

So different faces CANNOT have identical vertex sets, which excludes the hemicube for example.


 * the definition of an abstract polytope is well-established in refereed scientific literature dating back almost 30 years. Personal feelings are worth expresing, but not very relevant to the content of the article. As it happens, there are definitions in the field that I don't like, too - eg, I feel the definition of "regular" is too strong. But I stick to the definitions when writing articles (for wikipedia or for J. XYZ Math... :-). mike40033 (talk) 11:59, 11 April 2008 (UTC)

This definition of CONNECTEDNESS probably should be in a separate article. However I do not have references to support it - can anyone provide these?


 * Various equivalent definitions of "strong connectedness" appear in virtually every article on abstract polytopes. However, I don't know any references that support the idea that the definition should be in a separate article... :-) What exactly did you mean, though? mike40033 (talk) 11:59, 11 April 2008 (UTC)

(4) Has anyone else noticed the remarkable fact that the Hasse diagram of any polytope is itself the graph of a polytope of 1 dimension higher? The latter has only tetragonal 2-faces. A triangle generates a cube, for example.


 * I can't recall whether this has been noticed or not. Perhaps check Schulte's Reguläre Inzidenzkomplexe (circa 1980)? mike40033 (talk) 11:59, 11 April 2008 (UTC)

Contributed by Stephen L Woolf (I will put all this in the article later, or create a new article, if someone can help with supporting references).

SLWoolf (talk) 13:48, 13 March 2008 (UTC)

Suggested Improvements to the Definition
The definition is, I think, basically good - but could be improved still further.

(a) Redundancy.  As I have stated earlier in the discussion, it is superfluous to specify that the Poset has a Rank Function. The Rank or Dimension of any member of the poset is easily defined as 1 less than the 'vertical' distance between it and the minimal element. This ensures that the faces have ranks from -1 to n.

(b) Rigour.  In the article, Axiom 2 is followed by the statement:

"Given faces F, G of P with F < G, the section G/F = {H | F < H < G} has rank G/F = rank G − rank F − 1."

While this is clearly true, is this supposed to be an axiom, a clarification, or a theorem? As it seems to follow from the axioms, it would be a simple theorem, and should be therefore be stated afterwards as such. (Proving that a section is indeed a polytope is trivial for axioms 1,2,4 - I am not sure about axiom 3.)


 * Fixed! Mike(333+(42)3=343+(32)3) 04:33, 30 October 2008 (UTC)
 * Much better - but can you prove that a section is properly connected? The other 3 axioms are easily shown. Then the claim that a section is a polytope will be justified. If this in not provable from the axioms, then the axioms, in my view, are insufficient. When I started trying to define abstract polytopes way back in 1970, my original axiom was that an n-polytope is a union of (n-1)-polytopes and I still wonder if we really have the best axiomatisation now, though I must concede the current definition is beautifully concise.SLWoolf (talk) 06:02, 30 October 2008 (UTC)
 * It's proven in the literature. See, eg, the second chapter of McMullen & Schulte's book (given in the reference list). It seems that abstract polytopes were an idea whose time had come about 3 decades ago. I independently re-invented them about 15 years too late when pursuing my postgrad degree... mike40033 (talk) 06:56, 30 October 2008 (UTC)


 * My childhood obsessions with both the London Tube map and astronomy/cosmology inevitably led me to the idea that the universe itself might have a discrete geometry. The search for a concept of dimension, helped along by truly inspiring university courses in logic, set theory, and the foundations of maths, soon led me into (abstract) polytope theory.SteveWoolf (talk) 06:51, 31 October 2008 (UTC)

(c) Equivalence. I have now been able to prove with 95% confidence that the article's connectedness Axiom (3) is equivalent to my definition given previously.

(d) Generality. Clearly, the (current) definition of abstract polytope is an extremely general one. More familiar polytopes, such as a triangle or cube, are clearly a special class to which the digon and hemicube (for example) do not belong.

To illustrate, if mathematics studied only integers without the reals ("decimals"), it would be missing a lot. But clearly at times we also want to look at the unique properties of just those reals which are integers, such as primality.

For this reason we should also define this special subclass, and I think the defining characteristic is as I stated previously - that no two i-faces (elements of the poset) can have the same set of vertices (0-faces) and, dually, nor can they be contained by the same set of (n-1)faces (facets). Or if you prefer this terminology: the set of vertices (0-faces) incident with a given i-face is unique for each i-face (-1≤i<n), and dually, for the set of facets ((n-1)-faces).

I previously suggested Normal as a name for these basic polytopes. Any other ideas?


 * I believe that in the literature they are called 'non-flat' polytopes. Mike(333+(42)3=343+(32)3) 04:33, 30 October 2008 (UTC)

SLWoolf (talk) 09:14, 26 October 2008 (UTC)

Keep in mind that wikipedia is for reporting the facts, not inventing them. Mike(333+(42)3=343+(32)3) 04:34, 30 October 2008 (UTC)

Thanks, see also my dialog with Steelpillow in "Johnson's Approach" earlier. Perhaps I should take inspiration from you and make my original ideas part of my username!


 * as you can see, I've changed it back to plain ole 40033. The complex formula looked too cluttered... mike40033 (talk) 08:19, 31 October 2008 (UTC)

SLWoolf (talk) 05:16, 30 October 2008 (UTC)

Stub class
This article is obviously not "stub" class anymore. Can someone familiar with the different classes please update this? mike40033 (talk) 11:46, 11 April 2008 (UTC)

I have changed it to START class.SLWoolf (talk) 09:19, 26 October 2008 (UTC)

Over my head: hemi-octahedron
Seeing an open link above, I added a stub hemi-octahedron with an image. It surprised me to see it has 3 vertices and 6 edges, so every vertex is in contact with ever other vertex by 2 edges! This makes my head spin a bit. "Normally" I'd imagine this can happen like as a spherical tiling with digons on two polar opposite vertices, like a trigonal hosohedron, {2,3} has 2 vertices and 3 edges. Anyway, it's 1-skeleton is a triangular multigraph I guess. Tom Ruen (talk) 23:18, 29 October 2008 (UTC)


 * P.S. Like the others, I don't see why a hemi-tetrahedron doesn't exist, with 2 vertices, 3 edges, and 2 triangular faces. I guess it has a further weirdness - loop edges from a vertex back to itself, although no worse than a henagonal dihedron, {1,2}, I think!


 * it's hard to see how a triangular face could have two or fewer vertices, and still be an abstract polygon. Mike(333+(42)3=343+(32)3) 04:35, 30 October 2008 (UTC)


 * Abstractly, I suspect that it may indeed be a valid polytope: I just sketched a Hasse diagrams which has the correct numbers of V, F and E, and is monal, dyadic and properly connected. It has three edges at each vertex, and every edge meets two vertices and two faces. It is self-dual. Like the other hemiregulars it has an Euler characteristic of 1 suggesting that it can be realized unfaithfully as a tiling of the real projective plane - or, embedded in Euclidean space, of the Boy surface or similar. But figuring out the tiling is giving me some trouble: the edges are not straight, for a start. In passing, the ordinary abstract tetrahedron can be realized unfaithfully as a kind of "hemi-stella octangula" - a double-covered tiling of the Boy surface. -- Cheers, Steelpillow (Talk) 10:56, 30 October 2008 (UTC)


 * Choose a face, and one of its vertices. How many edges are between them? If you have an abstract polytope, the answer must be 2, and I'd love to see it. Otherwise, you probably have this : V = {a,b}, E = {1,2,3}, F = {C, D} with a < 1,2,3; b < 1,2,3; 1,2,3 < C and 1,2,3 < D. Am I right? mike40033 (talk) 00:00, 31 October 2008 (UTC)


 * You are right. That's just what I have and it is not monal after all (i.e. for a given triangular face, one vertex has been counted twice). -- Cheers, Steelpillow (Talk) 13:15, 1 November 2008 (UTC)

Improved Definition?
I offer a slightly more formal definition without the redundancy of requiring the rank function within the definition.

An abstract n-polytope is a partially-ordered set (poset) P, whose elements we call faces, satisfying:


 * (P1) P is bounded, i.e. it has a unique minimal face and a unique maximal face (usually, but not necessarily distinct)


 * (P2) Every maximal chain (which we call a flag) has exactly n + 2 faces.


 * (P3) It is strongly connected, that is, any flag can be "changed" into any other by "changing" just one face at a time.


 * (P4) If faces F, G, H satisfy F&raquo;G&raquo;H, then there is precisely one other face G'≠G such that F&raquo;G'&raquo;H, where &raquo; means "covers" or "succeeds": x covers z if x>z and no y satisfies x>y>z.

...

The rank or dimension of a face F is z-2, where z is the number of elements in a "vertical" chain from the minimal element to F. A face of dimension k is a k-face.

Yes/No/Too trivial and cosmetic to discuss further? If no further response I will assume and accept the latter, and move on to more crucial topics. SteveWoolf (talk) 14:50, 31 October 2008 (UTC)

Spherical vs Toroidal polytopes
Did Mike mean to say Genus 1 as opposed to rank 1? Rank in abstract polytope theory is synonymous with dimension, is it not? (Maybe us geometers should use dimension only, and leave rank to the order-theorists).


 * no, I meant rank. So I define the line segment to be spherical, and then go inductively on the dimension (rank). Ooops! I just realised this is not going to work. You'd need to also define (finite) polygons as spherical. In fact, now my memory is reminding me that I actually did define 'spherical polytope' this way in an article once (submitted, but no news from the referees yet). So this gives..... (see below.....)mike40033 (talk) 08:27, 31 October 2008 (UTC)

Previously, I tried to exclude toroidal polytopes from my definition. I accept them now, but we need to define what spherical means. I had something along the lines of


 * P is spherical if it is not the union of two polytopes that intersect at two disjoint facets

- but whether this is useful I'm not sure.


 * this would fail to exclude, eg, the tesselation of the plane with squares, I think... mike40033 (talk) 08:27, 31 October 2008 (UTC)

I think it is both interesting and crucial to point out that the 4-cube and the 3-dimensional toroid made up of 4 cubic cells in a ring both have the same graph, a fact that puzzled me for a long time. I originally tried to define an abstract polytope as just a graph (i.e. vertices and edges only) and finally realised that an enumeration of all faces was required to uniquely determine a polytope. SteveWoolf (talk) 07:17, 31 October 2008 (UTC)


 * As do the 11-cell, and the 10-simplex... or the hemi-dodecahedron and the 9-simplex... or the hemi-cube and the tetrahedron... mike40033 (talk) 08:27, 31 October 2008 (UTC)

Topological definitions
The usual topological definitions are based on the genus of a polytope, both orientable g and non-orientable k. These are related to the Euler characteristic &chi; = V - E + F - C ..., where V, E, F, C etc. are the number of vertices, edges, faces, cells, and so on.

I am not sure how it works for higher polytopes, but for polyhedra we get:


 * Spherical: &chi; = 2, g = 0,
 * Toroidal: &chi; = 0, g = 1,
 * Klein bottle: &chi; = 0, k = 2
 * Boy surface: &chi; = 1, k = 1
 * For higher genus non-orientable toroids, there is a trade-off between g and k.

It is trivial to find &chi; for any abstract polytope. If rank (dimensionality) = 3 and &chi; = 2 then we have a spherical polyhedron. I do not know if the equivalent works in higher dimensions.

Establishing the orientability of an abstract polytope is a trickier problem, and I am not sure if it has been solved. 13:38, 1 November 2008 (UTC)

Spherical polytopes (reprise)
Now for an abstract definition of a spherical polytope

A polytope is spherical if and only if one of the following hold :


 * it has rank 1
 * it is finite, and has rank 2, OR
 * it is finite and universal, with spherical facets and vertex figures.

mike40033 (talk) 08:28, 31 October 2008 (UTC)

Tea-Break
Or meta-meta-article.

It's great that there is so much interest in these topics, which reinforces my feeling that this subject might soon become dramatically more important a la Cartesian Geometry or Topology.

I'd like to make a short list of "our" current points and outstanding questions, but before that....

This discussion page is getting long, disordered, outdated - not to mention full of things I for one should never have written! Can we archive most of it and start fresh? Or just keep the good stuff if we can agree on that - I'm sure we could. (We can start axing right here...after discussion in a meta-meta-meta-article.) 15:12, 31 October 2008 (UTC) —Preceding unsigned comment added by SteveWoolf (talk • contribs)


 * Excellent idea, if you have got the time. -- Cheers, Steelpillow (Talk) 13:49, 1 November 2008 (UTC)